copyright © 2006 axioma, inc. robust portfolio construction sebastian ceria chief executive officer...

Copyright © 2006 Axioma, Inc.

Robust Portfolio Construction

Sebastian CeriaChief Executive Officer

Axioma, [email protected]

Presentation to Workshop on Mixed Integer Programming

University of Miami

June 5-8, 2006

Copyright © 2006 Axioma, Inc. 2

Glossary

Assets (n)Investable securities (U), typically stocks (equities)

Portfolio TradesHoldings: Initial (h), final (x) (holdings are represented in % or dollars) (x-h)Long holdings: (i : xi > 0)Short holdings: (i: xi < 0)

BenchmarkA market portfolio: S&P 500, Russell 1000 (typically market-cap weighted) (b)

BudgetThe total amount invested (B)

Expected Returns (Expected Active Returns)A vector (α) of expectations of return (in percent), expected return of a portfolio α’x (α’(x-b))

Covariance of ReturnsA matrix (Q) representing the forecasted covariances of returns

Predicted Risk of a portfolio Predicted Tracking Errorx’Qx (x-b)’Q(x-b)


Roadmap of the Quant Process

Estimation

Process

Estimation

Process

Historical

Data

Historical

Data

Forecasted

Expected

Returns

Forecasted

Expected

Returns

Hot TipsHot Tips

Fundamental

Analysis

Fundamental

Analysis

Sun SpotsSun Spots

MV

Optimization

MV

Optimization

Optimized

Portfolio

Optimized

Portfolio

Parameter EstimationParameter Estimation Portfolio Construction Portfolio Construction

Forecasted

Risk

Forecasted

Risk

Business

Rules

Business

Rules


Uix

RbxQbx

Bx

x

i

t

Uii

t

,0

)()(

st.

Max

Mathematical Models (MVO)

Active Management

Expected Returns

Budget

No Shorting

Tracking Error

)()( bxQbx t Risk Aversion


Why Don’t Practitioners Use MVO Extensively?

Naïve portfolio rules, such as equal weighting, can outperform traditional MVO (Jobson and Korkie)

Optimal portfolios from MVO are not necessarily well diversified (Jorion) or intuitive (Several authors)

MV Optimizers have the “Error Maximization Property” . MVO will tend to overweight assets with positive estimation error and underweight assets with negative estimation error (Several authors)

Unbiased risk and expected return estimators still lead to a biased estimate of the efficiency frontier (Several authors)

Portfolio Managers spend most of their time “cleaning up” the optimal portfolio provided by MVO


Criticisms of MVO

Literature Review: There is an extensive literature that studies the effects of estimation error in classical mean variance optimization

Jobson and Korkie, “Putting Markowitz Theory to Work”, JPM, 1981 (and related work)

Jorion. “International Portfolio Diversification with Estimation Risk, Journal of Business, 1985 (and related work)

Chopra and Ziemba, The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice”, JPM, 1993

Broadie, “Computing Efficient Frontiers Using Estimated Parameters”, Annals of OR, 1993

Michaud, “The Markowitz Optimization Enigma: Are Optimized Portfolios Optimal?”, FAJ 1989 and “Efficient Asset Management”, Oxford Univ Press, 1998


How do Practitioners “fix” MVO? (Extensions)

Simple (Linear):Initial holdings (h)

Transaction variables (t = |x h|)

Limits on holdings/trades (x u, t v)

Industry/Sector Holdings (iSxi c)

Active Holdings, Industry/Sector Active Holdings (|x b| u , iS|xi bi | c)

Limits on Turnover, Trading, Buys/Sells (iSti c)

Complex (Linear-Quadratic):Long/Short Portfolios (eliminate x 0)

Multiple Risk constraints xtQ x c

Multiple Tracking Error constraints (x b)tQ (x b)) c

Soft constraints/objectivesAdd “slack” to the constraint iSxi = c is modified to iSxi + s = c

Add S2 to the objective as a penalty term



Transaction cost modelsLinear: (add to the objective iUiti)Piecewise Linear (convex)

Market impact models Quadratic (add to the objective iUiti

2)Piecewise Linear

Risk Models (Common factors)Exploit mathematical structure of factor modelsFactor related constraints/objectives

Q matrix is split into specific and factor riskQ = ETRE + S2

E = Exposure matrix, R = Factor Covariance Matrix (dense), S = Specific risk matrix (diagonal)Typically 50-100 factors are used



Fixed Transaction Costsif ti > 0 then add ci to the objective Fixed Costs

Threshold Transactionsti = 0 or ti c

Threshold Holdingsxi = 0 or xi c

Maximum number of Transactions/Holdings|{i | xi 0 }| c

Round Lots on Transactionsti {kc | k = 1,2,…}

If … then … else conditionsif {linear expression} then {linear expression} else {linear expression}


Practical Solution of MVO with Extensions

In its general form, the problem is a Quadratic Objective, Quadratic and Linearly Constrained Mixed-Integer (Disjunctive) Programming Program

Even though this problem is “hard” we can “solve” efficiently most practical instances in a few seconds

Our algorithm includesPreprocessing: Problem reduction and formulation improvement

Strong relaxation: Reformulation and strengthening techniques

Heuristics: Used to find feasible solutions (portfolios) fastRelax-and-Fix

Diving

Branching: Specialized branching to exploit the structureCardinality constraints

Semi-continuous variables

Disjunctive statements


Is this Enough?

Assume you are indifferent (from a risk perspective) between the two assets, how would you weigh these two assets in the optimal portfolio?

Would you make the same choice if you knew the distribution of expected returns?

Expected Return

XYZ 15.0%

UVW 14.5%

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Expected Return (%)

UVW

XYZ


Three Asset Example: shorting allowed, budget constraint

Expected returns and standard deviations (correlations = 20%)

Optimal weights

1 2

Asset 1 7.15% 7.16% 20%

Asset 2 7.16% 7.15% 24%

Asset 3 7.00% 7.00% 28%

Portfolio E Portfolio F

Asset 1 67.18% 67.26%

Asset 2 43.10% 43.01%

Asset 3 -10.28% -10.28%

Improving Stability


Graphical Representation


Three Asset Example: no shorting, budget constraint


Optimal weights

1 2

Asset 1 7.15% 7.16% 20%

Asset 2 7.16% 7.15% 24%

Asset 3 7.00% 7.00% 28%

Portfolio A Portfolio B

Asset 1 38.1% 84.3%

Asset 2 61.9% 15.7%

Asset 3 0.0% 0.0%

Improving Stability II


Constraints Creates Instability


Stability Experiment on the Dow 30

Instability due to changes in Expected Returns:

Use expected returns and covariance from Idzorek (2002) for Dow 30

Randomly generate 10,000 expected return estimate vectors from a normal distribution with mean equal to the expected return and std equal to 0.1% of the of the std of return of the corresponding asset

Run 10,000 traditional MVO and record the weights of the resulting portfolios – Use a fixed risk aversion coefficient

0%

5%

10%

15%

20%

25%

Figure 1: Range of Expected Returns used in MV Optimization

0%

5%

10%

15%

20%

25%

Figure 2: Range of Asset Weights for MV Optimization


Efficiency Frontiers and Classical MVO

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550

Risk

Ret

urn

Estimation Error Generates Inefficiencies

How does the True Efficiency Frontier differ from the Actual Frontier?

Estimated Frontier

Efficiency Frontier computed using the estimated expected

returns

True Frontier

Efficiency Frontier computed using the

true expected returns

Actual Frontier

Return for the portfolios in the

Estimated Frontier using the true

expected returns

* See Broadie (1993) for a detailed discussion of estimated frontiers

1. Take a Portfolio on the Estimated Frontier

2. Apply the TRUE expected returns

3. Measure its REALIZED expected return and graph accordingly

Actual Frontier


One Possible Solution

A byproduct of the estimation process is a distribution of estimated expected returns, and not a point forecast

One option is to sample from the distribution and average the resulting portfolios: resampling

Averaged

Optimized

Portfolio

Averaged

Optimized

Portfolio

Parameter Estimation Portfolio Construction

MV

OptimizationEstimation

Process

Estimation

Process

Historical

Data

Historical

Data

Hot TipsHot Tips

Fundamental

Analysis

Fundamental

Analysis

Sun SpotsSun Spots



Integrating the Estimation Process and Robust MVO

Robust MVO uses explicitly the distribution of forecasted expected returns (estimation error) to find a robust portfolio in ONE step

Estimation

Error

Estimation

Error

Robust MV

OptimizationRobust MV

Optimization

Robust

Optimized

Portfolio

Robust

Optimized

Portfolio

Parameter Estimation Portfolio Construction Parameter Estimation Portfolio Construction

Estimation

Process

Estimation

Process

Historical

Data

Historical

Data

Hot TipsHot Tips

Fundamental

Analysis

Fundamental

Analysis

Sun SpotsSun Spots



The Proposed Solution: Robust MVO

Robust Mean-Variance Optimization relies on “Robust Optimization” to solve the Portfolio Construction Problem

What is Robust Optimization?An optimization process that incorporates uncertainties of the inputs into a deterministic frameworkIt explicitly considers estimation error within the optimization processIt was developed independently by Ben-Tal and Nemirovski. Initial applications were in the area of engineering

What are the advantages of using Robust MVO?Recognize that there are errors in the estimation process and directly “exploit” that knowledge Address practical portfolio construction constraints directly and explicitlySolve the Robust MVO problem “efficiently” in “roughly” the same time as ONE classical mean variance optimization problem

How do we solve Robust MVO problems?The Robust MVO problem can be formulated as a “Second Order Cone Programming Problem” (Linear Programming over second-order cones)Interior Point Algorithms are used to optimize SOCPs

“When solving for the efficient

portfolios, the differences in

precision of the estimates should

be explicitly incorporated into

the analysis”

H. Markowitz


Robust Optimization Background

Literature Review: Even though Robust Optimization is relatively a new discipline, there is already an extensive literature in the subject for portfolio management

A. Ben-Tal and A.S. Nemirovski, "Robust convex optimization", Math. Operations Research, 1998

A. Ben-Tal and A.S. Nemirovski, "Robust solutions to uncertain linear programs", Operations Research Letters, 1999

L. El Ghaoui, F. Oustry, and H. Lebret, "Robust solutions to uncertain semidefinite programs", SIAM J. of Optimization, 1999

M. Lobo and S. Boyd, “The Worst Case Risk of a Portfolio”, 1999

R. Tütüncü and M. Koenig, “Robust Asset Allocation”, 2002

D. Goldfarb and G. Iyengar, “Robust Portfolio Selection Problems”, Math of OR, 2003

L. Garlappi, R. Uppal, and T. Wang, “Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach”, 2004

D. Bertsimas and M. Sim, Robust Discrete Optimization and Downside Risk Measures, 2005


How do we maximize Robust Expected Returns?

Assume the vector of expected returns ~ N(*,)

Define an elliptical confidence region around the vector of estimated expected returns * as {: ( - *)T -1( - *) ≤ k2}

(if the errors are normally distributed k2 comes from the chi-squared distribution)

Robust Objective: The optimization problem is defined as:

Max (Min E(return)) = Maxx Min(*) E(x)

And (*) is the region around * that will be taken into account as potential errors in the estimates of expected returns

Maximizing Robust Expected Returns

“A robust objective adjusts estimated expected returns to counter the (negative) effect that optimization has on the estimation errors that are present in the estimated

expected returns”


Intuitive Derivation of Robust MVO

Efficiency Frontiers and Classical MVO

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550

Risk

Ret

urn

Minimize

DistanceEstimated Frontier

Efficiency Frontier computed using the estimated expected

returns

True Frontier



Actual Frontier



expected returns


Mathematical Formulation of Robust MVO

Estimation Error “Aversion”

Additional term that “corrects” for estimation error

Second Order Cone

Programming Problem

Additional term that “corrects” for

risk

Maximize expected return

aversionrisk

aversionerror estimation

returns ofmatrix covariance

returns expected estimated ofmatrix covariance

returns expected of vector

0

)'(max

*

* 2

1

k

Q

x

Bxest

Qxxxkx

t

Risk “Aversion”


Impact of the Proposed Solution

Measuring the improvement: How do we know we are doing better?Reducing overestimation/underestimation

Compute the difference between the estimated and actual efficient frontiers

Improving stability

Compute a “measure” of the variability of weights given the variability in expected returns

Improving the “information transfer coefficient”, a measure that explains how information is being transferred

Measure of how efficiently an investment process is able to use the forecasting information it generates


Intuition behind Robust MVO: Simple Example

Three Asset Example: no shorting, budget constraint


Optimal weights

1 (A) 2 (B)

Asset 1 7.15% 7.16% 20%

Asset 2 7.16% 7.15% 24%

Asset 3 7.00% 7.00% 28%

High Aversion Medium Aversion Low Aversion

A B A B A B

Asset 1 35.26% 35.68% 43.38% 45.54% 47.36% 52.65%

Asset 2 35.69% 35.27% 45.55% 43.39% 52.64% 47.35%

Asset 3 29.05% 29.05% 11.07% 11.07% 0.0% 0.0%


Efficiency Frontiers and Robust MVO

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500

Risk

Ret

urn

Reducing Overestimation/Underestimation

Estimated Robust Frontier

Robust Efficiency Frontier computed

using the estimated expected returns

and the estimation error

True Frontier



Actual Robust Frontier



expected returns


Improving Optimal Portfolio Stability and Intuition

0%

5%

10%

15%

20%

25%

Figure 2: Range of Asset Weights for MV Optimization

0%

5%

10%

15%

20%

25%

Figure 2: Range of Asset Weights for Robust Optimization

Resultant asset weights using error maximized optimization vs. Robust MVO for the prior Dow 30 example

Lower ranges in asset weights

Less variability across asset weights


Improving the Transfer of Information

Define a measure that allows us to determine how the information contained in the estimated expected returns is being “transferred” to the portfolio via the optimizer.

We use the correlation of the implied alphas to the true alphas


Summary

Robust MVO incorporates information about the estimation process directly into the optimization problem

Robust MVO takes into account those estimation errors when computing the portfolio that maximizes utility

Robust MVO improves performance through less trading and better use of the information at hand

Robust MVO is a one-pass procedure which is efficiently implemented through an SOCP algorithm, it allows for the addition of other constraints

Robust MVO naturally diversifies, even in the absence of a risk model


Improving Performance

Start with a predefined covariance matrix and a vector of expected returns

Randomly generate a time-series of returns for each asset, with the appropriate correlation

To get estimated expected returns, we use “simplistic” estimators that average returns over previous periods with some weight assigned to current period (to put some look-ahead bias)

To get the distribution of errors for estimated expected return we compute the covariance matrix over the same time periods. We scale the resulting matrix by a factor 1/v

Sharpe Ratio is computed once, at the end of each run by taking the actual returns divided by their STD

Each back-test is run 100 times

For each time-period, we solve a problem with the following constraintsRisk constraint using the “true” covariance matrix (10% dollar-neutral)

Long/Only – Active No shorting

Turnover constraint at 7.5% (each way)

Asset bounds: 15% L/S [-15%,15%]


Simulated Back-Test Results (Dollar-Neutral)

MVO Robust MVO

Confidence Level

Ann. Return*

Sharpe Ratio*

Ann.

Return*

Sharpe Ratio*

% Imp.

Sharpe Ratio

% Improved Returns

High 0.59% 0.1289 1.09% 0.1455 13% 73%

Medium 0.59% 0.1289 1.95% 0.1767 37% 70%

Low 0.59% 0.1289 2.95% 0.2260 75% 76%

Very Low 0.59% 0.1289 3.77% 0.2812 118% 78%

High 1.49% 0.1657 2.02% 0.1847 11% 74%

Medium 1.49% 0.1657 3.11% 0.2307 39% 72%

Low 1.49% 0.1657 4.23% 0.2927 77% 80%

Very Low 1.49% 0.1657 5.21% 0.3653 120% 81%

* Averages computed over 100 runs of the back-tests

Low

Info

Hig

h In

fo


Simulated Back-Test Results (Active Management 5%)

MVO Robust MVO

Confidence Level

Ann. Return*

Inf. Ratio*

Ann.

Return*

Inf. Ratio*

% Imp.

Inf. Ratio

% Improved Returns

High 4.22% 0.5524 4.43% 0.5840 6% 77%

Medium 4.22% 0.5524 4.80% 0.6379 15% 77%

Low 4.22% 0.5524 5.00% 0.6641 20% 81%

Very Low 4.22% 0.5524 5.02% 0.6607 20% 70%

High 6.45% 0.8022 6.72% 0.8465 6% 79%

Medium 6.45% 0.8022 7.17% 0.9234 15% 81%

Low 6.45% 0.8022 7.36% 0.9594 20% 80%

Very Low 6.45% 0.8022 7.33% 0.9555 19% 74%

* Averages computed over 100 runs of the back-tests

Low

Info

Hig

h In

fo

copyright © 2006 axioma, inc. robust portfolio construction sebastian ceria chief executive officer...

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